The formula is a geometric series. What it shows is that monthly churn outweighs the cost of capital, and why an open-access line can be worth more than a retail one.
A fibre subscriber's lifetime value is the monthly margin, ARPU less the cost to serve, times the chance the customer is still there each month, discounted: LTV = margin × (1 − churn) / (monthly discount rate + churn). At $70 of ARPU, $18 of cost to serve, 1.3 per cent monthly churn and a 10 per cent discount rate, a retail subscriber is worth $2,447, or $1,747 after the $700 connection. The shortcut of margin divided by churn says $4,000 and overstates it by 63 per cent.
Worked in full in The Digital Infrastructure Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The inputs are those of Oakmere Fiber, the fictional fibre-to-the-home network in The Digital Infrastructure Investor, which the book runs both as a retail operator and as an open-access wholesaler. Margins are held flat in real terms; the case's ARPU and opex escalators are left out to keep the formula visible.
| Input | Retail | Open access |
|---|---|---|
| ARPU per month | 70 | 40 |
| Cost to serve per subscriber per month | 18 | 5 |
| Monthly margin | 52 | 35 |
| Churn per month | 1.3% | 0.5% |
| Connection cost, first subscriber at a home | 700 | 700 |
| Annual discount rate (the fund's IRR target) | 10% | 10% |
Churn is monthly, so the discount rate must be monthly too, compounded, not divided by twelve.
Monthly discount rate = (1 + 10%)1/12 − 1 = 0.7974%
Annual churn = 1 − (1 − 1.3%)12 = 14.5%, not 12 × 1.3% = 15.6%
Expected life = 1 / churn = 1 / 1.3% = 76.9 months, about 6.4 years
The monthly to annual conversion is worked in more detail in how to convert monthly churn to annual churn.
A subscriber still connected at the end of month t pays the margin for that month. The probability of that is (1 − churn)t, and the payment is discounted by (1 + r)t. The sum of that geometric series has a closed form.
LTV = margin × (1 − c) / (r + c)
= $52 × (1 − 1.3%) / (0.7974% + 1.3%) = $52 × 0.987 / 2.0974% = $2,447
Net of connection: $2,447 − $700 = $1,747; undiscounted payback of the connection: $700 / $52 = 13.5 months
Excel: =Margin*(1-Churn)/(((1+Disc)^(1/12)-1)+Churn)-Connection
Churn and the discount rate sit side by side in the denominator, which is the useful way to read the formula: at these inputs churn of 1.3 per cent a month weighs more than the cost of capital, 0.7974 per cent. Halving churn does more for value than halving the discount rate.
LTV, open access = $35 × (1 − 0.5%) / (0.7974% + 0.5%) = $2,684, or $1,984 after connection
The wholesale line earns a margin 1.49 times smaller and is still worth 9.7 per cent more. The reason is churn. When a customer on an open-access network switches retailer, the wholesale line usually stays lit; it is lost only when the household leaves fibre or moves out. At 0.5 per cent a month, 5.8 per cent a year, the expected life is 200 months against 76.9 for retail.
Per-subscriber value is not the network's return. The book's open-access variant earns 7.55 per cent against 10.34 for retail at network level, because penetration, overhead and the exit valuation also move. A network sold on a multiple of EBITDA is paid for its margin, and the multiple does not see churn at all.
| Monthly churn | Annual churn | Expected life, months | LTV | Net of $700 |
|---|---|---|---|---|
| 0.5% | 5.8% | 200 | 3,988 | 3,288 |
| 1.0% | 11.4% | 100 | 2,864 | 2,164 |
| 1.3% | 14.5% | 77 | 2,447 | 1,747 |
| 1.5% | 16.6% | 67 | 2,229 | 1,529 |
| 2.0% | 21.5% | 50 | 1,822 | 1,122 |
Value is not linear in churn. Moving from 1.3 to 0.5 per cent adds $1,541; moving from 1.3 to 2.0 per cent takes away $625. A connection stops paying for itself only at 6.17 per cent a month, 53.4 per cent a year, so the drop is rarely the problem. The problem is that every point of churn sits next to a discount rate below one per cent a month and dominates it.
Margin divided by churn: $52 / 1.3% = $4,000. It ignores the time value of money, and on a 77-month life that is a 63 per cent overstatement. The second mistake is fibre-specific: treating the $700 connection as a pure acquisition cost lost with the customer. The drop outlives the subscriber. When the home is re-let to a new customer, the case charges only a $200 reconnection, so the next subscriber at the same address is worth $2,247 net, and the first connection is closer to network capex than to a marketing cost. That is why penetration, not subscriber LTV, is the number the network is underwritten on, as in the penetration an FTTH network needs to earn 10 per cent.
The vintage-by-vintage model behind these inputs, with churn, reconnections and both operating models, is in the free workbook for this case.
There is no single benchmark, and any figure should be treated as illustrative. What matters is churn against the discount rate: at 10 per cent a year the monthly rate is 0.7974 per cent, so a retail network at 1.3 per cent monthly churn, 14.5 per cent a year, loses value faster to churn than to time. Cutting churn to 1.0 per cent lifts a $52-margin subscriber from $2,447 to $2,864.
Because switching retailer does not usually disconnect the wholesale line. In the illustrative case, open access earns $35 a month against $52 for retail, but churns at 0.5 per cent a month instead of 1.3. Its lifetime value is $2,684 against $2,447, 9.7 per cent more, even though the margin is 1.49 times smaller.
Deduct it for the first subscriber at a home, then remember that the drop survives churn. At $700 the first subscriber nets $1,747. A later subscriber at the same address costs only a $200 reconnection in the case, so is worth $2,247 net. The connection behaves more like network capex than like an acquisition cost.
This article is one calculation from The Digital Infrastructure Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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